Example for Basic Blocks

Transcription

C-9
Basic Blocks
Basic Block (dt. Grundblock):
Maximal sequence of instructions that can be entered only at the first of them and
exited only from the last of them.
Begin of a basic block:
• procedure entry
• target of a branch
• instruction after a branch or return
Call instructions:
are usually not considered as a branch
Local optimization
considers the context of one single basic block (or part of it) at a time.
© 2000 bei Prof. Dr. Uwe Kastens
Global optimization:
Basic blocks are the nodes of control-flow graphs.
Vorlesung Übersetzer II SS 2000 / Folie 09
Ziele:
Understand the notion of basic blocks
in der Vorlesung:
The topics on the slide are explained. Examples are given.
• The definition is explained.
• The construction is explained using the example of C-10.
• The consequences of having calls in a basic block are discussed.
Verständnisfragen:
• Explain the decomposition of intermediate code into basic blocks for C-10 and for further examples.
C-10
Example for Basic Blocks
A C function that computes Fibonacci numbers:
Intermediate code with basic blocks:
[Muchnick, p. 170]
int fib (int m)
{ int f0 = 0, f1 = 1, f2, i;
if (m <= 1)
return m;
else
{ for(i=2; i<=m; i++)
{ f2 = f0 + f1;
f0 = f1;
f1 = f2;
}
return f2;
} }
1
2
3
4
receive m
f0 <- 0
f1 <- 1
if m <= 1 goto L3
B1
5
i <- 2
B3
6 L1: if i <= m goto L2
B4
7
B5
return f2
8 L2: f2 <- f0 + f1
9
f0 <- f1
10
f1 <- f2
11
i <- i + 1
12
goto L1
B6
13 L3: return m
B2
Ziele:
Example for the construction of basic blocks
in der Vorlesung:
The decomposition into basic blocks is explained according C-9 using the example.
C-11
Control-Flow Graph
A control-flow graph (dt. Ablaufgraph) represents the control structure of a procedure.
Nodes:
basic blocks and 2 unique nodes entry and exit.
Edges:
branches from the end of one basic block to the beginning of a basic block.
Fundamental data structure for
• structural transformations
• data-flow analysis
• code motion
Graph traversal: depth-first search (DFS), breadth-first search (BFS)
Ziele:
Understand the notion of control-flow graphs
in der Vorlesung:
Examples are given.
• The definition is explained.
• The example of C-12 is explained.
• The representation of loops in control-flow graphs is compared to source language representation.
• Algorithms that recognize loops in control-flow graphs are presented at the end of this chapter.
Verständnisfragen:
• Why is the loop structure of source programs not preserved on the level of intermediate languages?
C-12
Example for a Control-flow Graph
Intermediate code with basic blocks:
Control-flow graph:
[Muchnick, p. 172]
1
2
3
4
receive m
f0 <- 0
f1 <- 1
if m <= 1 goto L3
B1
5
i <- 2
B3
entry
B1
6 L1: if i <= m goto L2
B4
7
B5
return f2
8 L2: f2 <- f0 + f1
9
f0 <- f1
10
f1 <- f2
11
i <- i + 1
12
goto L1
B6
13 L3: return m
B2
B2
B3
B4
B5
exit
Ziele:
Example for a control-flow graph
in der Vorlesung:
The control-flow graph represents the basic blocks and their branches, as defined in C-11.
Verständnisfragen:
B6
C-13
Data-Flow Analysis
Data-flow analysis (DFA) provides information about how the execution of a program may
manipulate its data.
Many different problems can be formulated as data-flow problems, for example:
• Which assignments to variable v may influence a use of v at a certain program position?
• Is a variable v used on any path from a program position p to the exit node?
• The values of which expressions are available at program position p?
Data-flow problems are stated in terms of
• paths through the control-flow graph and
• properties of basic blocks.
Data-flow analysis provides information for global optimization.
Data-flow analysis does not know
• input values provided at run-time,
• branches taken at run-time.
Its results are to be interpreted pessimistic.
Ziele:
Goals and ability of data-flow analysis
in der Vorlesung:
• The topics on the slide are explained.
• Examples for the use of DFA information are given.
• Examples for pessimistic information are given.
nachlesen:
Kastens / Übersetzerbau, Abschnitt 8.2.4
Verständnisfragen:
• What’s wrong about optimistic information?
• Why can pessimistic information be useful?
C-14
Data-Flow Equations
A data-flow problem is stated as a system of equations for a control-flow graph.
System of Equations for forward problems (propagate information along control-flow edges):
2 equations for each basic block:
Out (B) = Gen (B) ∪ (In (B) - Kill (B))
In (B) =
Example Reaching definitions:
A definiton d of a variable v reaches the
beginning of a block B if there is a path
from d to B on which v is not assigned
again.
This problem is stated over sets of
the assignments in a procedure:
Θ
Out (h)
h ∈pred(B)
B
.
pred (B) . (In - Kill) ∪ Gen = Out
.
In, Out, Gen, Kill
In, Out
variables of the system of equations for each block
Gen, Kill a pair of constant sets that characterize a block w.r.t. the DFA problem
Θ meet operator, here: Θ = ∪ union of the Out sets of the predecessor blocks.
Ziele:
A DFA problem is modeled by a system of equations
in der Vorlesung:
• The equation pattern is explained.
• Equations are defined over sets.
• In this example: sets of assignment statements at certain program positions.
• The meet operator being the union operator is correlated to "there is a path" in the problem statement.
• Note: In this context a "definition of a variable" means an "assignment of a variable".
nachlesen:
Verständnisfragen:
• Explain the meaning of In(B)= {d1: x=5, d4: x=7, d6: y=a+1} for a particular block B.
.
.
.
C-15
Specification of a DFA Problem
Specification of reaching definitions:
• Description: A definiton d of a variable v reaches the beginning of a block B if there is a path
from d to B on which v is not assigned again.
• It is a forward problem.
• The meet operator is union.
• The optimization information in the sets are assignments at certain program positions.
• Gen (B): contains all definitions d: v = e; in B, such that v is not defined after d in B.
• Kill (B): contains all definitions d: v = e; in blocks different from B,
such that B has a definition of v.
Variants of DFA problems:
• forward problem ; backward problem () (cf. C-17)
• union problem: Θ = ∪; „there is a path“; solution: minimal sets that solve the equations
intersect problem: Θ = ∩; „for all paths“; solution: maximal sets that solve the equations
• optimization information: sets of certain statements, of variables, of expressions.
Further classes of DFA problems over general lattices instead of sets, not considered here.
Ziele:
Specify a DFA problem systematically
in der Vorlesung:
• The items that characterize a DFA problem are explained.
• The definition of Gen and Kill is explained.
• The variants of DFA problems are summarized.
nachlesen:
Verständnisfragen:
• Why does this definition of Gen and Kill serves the purpose of the description in the first item?
• Explain the relation of the meet operator, the paths in the graph, and the maximal/minimal solutions.
C-16
Example Reaching Definitions
B1
Gen (B):
contains all definitions d: v = e;
in B, such that v is not defined
after d in B.
Kill (B):
contains all definitions d: v = e;
in blocks different from B,
such that B has a definition of v.
d1 : a :=
d2 : b :=
d3 : c :=
B2
B3
d4 : b :=
B4
B5
d5 : c :=
d6 : b :=
d7 : c :=
d8 : a :=
Gen
Kill
In
Out
B1
d1, d2, d3
d4, d5, d6, d7, d8
∅
d1, d2, d3
B2
d4
d2, d6
d1, d2, d3
d1, d3, d4
B3
d5
d3, d7
d1, d2, d3, d6, d7
d1, d2, d5, d6
B4
d6, d7
d2, d3, d4, d5
d1, d2, d5, d6
d1, d6, d7
B5
d8
d1
d1, d2, d3, d4, d5, d6
d2, d3, d4, d5, d6, d8
Ziele:
Understand the meaning of DFA sets
in der Vorlesung:
• The example for C-15 is explained.
nachlesen:
Verständnisfragen:
• Check that the In and Out sets solve the equations for the CFG.
• How can you argue that the solution is minimal?
• Add some elements to the solution such that it still solves the equations. Explain what such non-maximal solutions
mean.
C-17
Backward Problems
System of Equations for backward problems
propagate information against control-flow edges:
2 equations for each basic block:
In (B)
= Gen (B) ∪ (Out (B) - Kill (B))
Out (B) =
Example Live variables:
• Description: Is variable v alive at a given
point p in the program, i. e. is there a path
from p to the exit where v is used but
not defined before the use?
• backward problem
• optimization information: sets of variables
Θ
In (h)
h ∈succ(B)
B
control-flow
.
.
.
In = Gen ∪ (Out - Kill)
.
. succ (B)
.
optimization information
• meet operator: Θ = ∪ union
• Gen (B): variables that are used in B, but not defined before they are used there.
• Kill (B): variables that are defined in B, but not used before they are defined there.
Ziele:
Symmetry of forward and backward schemes
in der Vorlesung:
• The equation pattern is explained.
• The DFA problem "live variables" is explained.
nachlesen:
Verständnisfragen:
• How do determine the live variables within a basic block?
C-18
Important Data-Flow Problems
• Reaching definitions: A definiton d of a variable v reaches the beginning of a block B if there
is a path from d to B on which v is not assigned again.
DFA variant: forward; union; set of assignments
Transformations: use-def-chains, constant propagation, loop invariant computations
• Live variables: Is variable v alive at a given point p in the program, i. e. there is a path from
p to the exit where v is used but not defined before the use.
DFA variant: backward; union; set of variables
Transformations: eliminate redundant assignments
• Available expressions: Is expression e computed on every path from the entry to a program
position p and none of its variables is defined after the last computation before p.
DFA variant: forward; intersect; set of expressions
Transformations: eliminate redundant computations
• Copy propagation: Is a copy assignment c: x = y redundant, i.e. on every path from c to
a use of x there is no assignment to y?
DFA variant: forward; intersect; set of copy assignments
Transformations: remove copy assignments and rename use
• Constant propagation: Has variable x at position p a known value, i.e. on every path from
the entry to p the last definition of x is an assignment of the same known value.
DFA variant: forward; combine function; vector of values
Transformations: substitution of variable uses by constants
Ziele:
Recognize the DFA problem scheme
in der Vorlesung:
• The DFA problems and their purpose are explained.
• The DFA classification is derived from the description.
• Examples are given.
• Problems like copy propagation oftem match to code that results from other optimizing transformations.
nachlesen:
Kastens / Übersetzerbau, Abschnitt 8.3
Verständnisfragen:
• Explain the classification of the DFA problems.
• Construct an example for each of the DFA problems.
C-19
Iterative Solution of Data-Flow Equations
Input:
Output:
the CFG; the sets Gen(B) and Kill(B) for each basic block B
the sets In(B) and Out(B)
Algorithm:
repeat
stable := true;
for all B ≠ entry
{*}
do begin
for all V ∈ pred(B) do
In(B):= In(B) Θ Out(V);
oldout:= Out(B);
Out(B):= Gen(B) ∪ (In(B)-Kill(B));
stable:= stable and Out(B)=oldout
end
until stable
Complexity:
Initialization
Union: empty sets
for all B do
begin
In(B):=∅;
Out(B):=Gen(B)
end;
Intersect: full sets
for all B do
begin
In(B) := U;
Out(B):=
Gen(B)∪
(U - Kill(B))
end;
O(n3) with n number of basic blocks
O(n2) if pred (B) ≤ k << n for all B
Ziele:
Understand the iterative DFA algorithm
in der Vorlesung:
• Initialization variants are explained.
• The algorithm is explained.
nachlesen:
Kastens / Übersetzerbau, Abschnitt 8.2.5, 8.2.6
Verständnisfragen:
• How is the initialization related to the size of the solution for the two variants union and intersect?
• Why does the algorithm terminate?
C-20
Variants of DFA Algorithms
Heuristic improvement:
Goal: propagate changes in the In and Out sets as fast as possible.
Technique: visit CFG nodes in topological order in the outer for-loop {*}.
Then the number of iterations of the outer repeat-loop is only determined
by back edges in the CFG
Algorithm for backward problems:
Exchange In and Out sets symmetrically in the algorithm of C-19.
The nodes should be visited in topological order as if the directions of edges were flipped.
Hierarchical algorithms, interval analysis:
Regions of the CFG are considered nodes of a CFG on a higher level.
That abstraction is recursively applied until a single root node is reached.
The Gen, Kill sets are combined in upward direction;
the In, Out sets are refined downward.
Ziele:
Overview on DFA algorithms
in der Vorlesung:
• The variants of the algorithm of C-19 is explained.
• The improvement is discussed.
• The idea of hierarchical approaches is explained.
nachlesen:
Kastens / Übersetzerbau, Abschnitt 8.2.5, 8.2.6
Verständnisfragen:
• For a backward problem the blocks could be considered in reversed topological order. Why is that not a good idea?

Example for Basic Blocks

Transcription

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